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F-algebra
Collection
context | $F$ in ${\bf C}\longrightarrow{\bf C}$ |
definiendum | $\langle A,\alpha\rangle$ in $\text{it}$ |
postulate | $\alpha:{\bf C}[FA,A]$ |
Discussion
Think types $\mathrm{a}$ and $\alpha$'s of type
type Algebra f a = f a -> a
Example
The following examples assume that ${\bf C}$ contains all the relevant ingredients (e.g. products).
- Addition of natural numbers is a binary relation: $+:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$. Hence $\langle \mathbb{N},+\rangle$ is an $F$-algebra for the endofunctor with object map $FX:=X\times X$.
- Group actions on $X$ are maps $m:G\times X\to X$, so consider $F_GX:=G\times X$. The first example is also an $F_\mathbb{N}$-algebra.
Reference
Wikipedia: F-algbera